Laminar flow reactors with

Kirelic/lranspon for Isothermal Laminar-Flow Reactor with no Axial Dispersion [See Shinohara and Christiansen (I974J for ilie non-isoihermul... 

Example 8.1 Find the mixing-cup average outlet concentration for an isothermal, first-order reaction with rate constant k that is occurring in a laminar flow reactor with a parabolic velocity profile as given by Equation (8.1). 

This integral is a special function related to the incomplete gamma function. The solution can be considered to be analytical even though the function may be unfamiliar. Figure 8.1 illustrates the behavior of Equation (8.8) as compared with CSTRs, PFRs, and laminar flow reactors with diffusion. 

Figure 8.1 includes a curve for laminar flow with 3>AtlR = 0.1. The performance of a laminar flow reactor with diffusion is intermediate between piston flow and laminar flow without diffusion, aVI = 0. Laminar flow reactors give better conversion than CSTRs, but do not generalize this result too far It is restricted to a parabolic velocity profile. Laminar velocity profiles exist that, in the absence of diffusion, give reactor performance far worse than a CSTR. 

Consider an isothermal, laminar flow reactor with a parabolic velocity profile. Suppose an elementary, second-order reaction of the form A -h B P with rate SR- = kab is occurring with kui 1=2. Assume aj = bi . Find Uoutlam for the following cases ... 

Kinetic/transport for Isothermal Laminar-Flow Reactor with Axial Dispersion under Transient Open-Loop Operation... 

The material balance on a laminar-flow reactor with negligible mass diffusion (discussed later in this chapter) is ... 

Compare this result with the exact analytical formula for the laminar flow reactor with a second-order reaction... 

The effects of the velocity profile are mitigated by molecular diffusion in the radial direction (i.e., the cross-sectional direction). Diffusion in the axial direction is negligible. Reactor performance is better than in a laminar flow reactor with no diffusion. If the radial diffusion is high enough, concentration gradients in the reactor cross section are eliminated, and reactor performance approaches that of piston flow. 

The laminar-flow reactor with segregation and negligible molecular diffusion of species has a residence-time distribution which is the direct result of the velocity profile in the direction of flow of elements within the reactor. To derive the mixing model of this reactor, let us start with the definition of the velocity profile. 

The integral in the second term is a form of the exponential integral, normally tabulated as Ei z) [M. Abramowitz and LA. Stegun, Handbook of Mathematical Functions, Dover, New York, NY, (1965)]. Further discussion of the ideal laminar-flow reactor with a first-order reaction is given by Cleland and Wilhelm [F.A. Cleland and R.H. Wilhelm, Amer. Inst. Chem. Eng. J., 2, 489 (1956)]. 

Compare the result of the integration of the segregated-flow model for first-order reaction in a laminar flow reactor with the result presented in Chapter 4. 

The derivative of a step change is a delta function, and f(t) = 8(t — t). Thus, a piston flow reactor is said to have a delta distribution of residence times. The variances for these ideal cases are = 1 for a CSTR and 0 = 0 for a PFR, which are extremes for well-designed reactors in turbulent flow. Poorly designed reactors and laminar flow reactors with little molecular diffusion can have 0 values greater than 1. 

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